Systems and methods for modeling neural architecture

ABSTRACT

Systems and methods are described herein for modeling neural architecture. Regions of interest of a brain of a subject can be identified based on image data characterizing the brain of the subject. the identified regions of interest can be mapped to a connectivity matrix. The connectivity matrix can be a weighted and undirected network. A multivariate transformation can be applied to the connectivity matrix to transform the connectivity matrix into a partial correlation matrix. The multivariate transformation can maintain a positive definite constraint for the connectivity matrix. The partial correlation matrix can be transformed into a neural model indicative of the connectivity matrix.

CROSS-REFERENCED TO RELATED APPLICATION

This application is a U.S. national stage entry of InternationalApplication No. PCT/US17/51343, filed on Sep. 13, 2017, entitled“SYSTEMS AND METHODS FOR MODELING NEURAL ARCHITECTURE,” which claims thebenefit of U.S. Provisional Application No. 62/393,949, filed on Sep.13, 2016, entitled “SYSTEMS AND METHODS FOR MODELING NEURALARCHITECTURE”, the contents of which are incorporated herein byreference.

GOVERNMENT LICENSE RIGHTS

This invention was made with government support under Grants NumberSES-1357622, SES-1357606, and SMA 1533500 awarded by the NationalScience Foundation. The government has certain rights in the invention.

TECHNICAL FIELD

This disclosure generally relates to systems and methods for modelingneural architecture. More specifically, this disclosure relates tosystems and methods for inferential testing, network modeling, andnetwork simulation of the neural architecture.

BACKGROUND

The brain is highly dynamic—billions of neurons across hundreds of brainregions interact to give rise to human cognition. Neural activity datacan be used to capture the degree of activity between regions of thebrain. The neural activity data can be collected from various inspectionmodalities. For example, functional magnetic resonance imaging (fMRI)images, diffusion tensor imaging (DTI) images, diffusion weightedimaging (DWI) images, electromyogram (EMG), electroencephalogram (EEG),positron emission tomography (PET), magnetoencephalography (MEG),Electrocorticography (ECoG), ultra-sound imaging, and their combinationcan be collected from brain matter of a patient. However, improvementsin the ability to gather neural activity data has outpaced the currentanalytical capabilities of neuroscience.

Currently, analytical tools are available for analyzing neural activitydata. However, existing analytical tools analyze neural activity datawithout substantially altering the neural activity data prior totesting. Moreover, existing analytic tools for analyzing the neuralactivity data are limited to the analyses of binary networks in whichconnections are either “on” (conventionally coded 1) or “off”(conventionally coded 0). Thus, existing tools for the analysis ofneural activity data fail to account for the weighted (valued) nature ofnetwork connections. In general, both descriptive and inferentialanalyses include determining a strength value between two regions of thebrain (e.g., a correlation strength between −1 and 1; or probability ofconnection between two regions), and generating a binary (on or off)connection based on a threshold. Thus, such analyses only considerconnections between regions of the brain that exceed the threshold.Thus, analysis of neural activity data, both descriptive andinferential, fail to effectively utilize information captured by theneural activity data on the relative strength of connections betweenregions of the brain. Moreover, analysis of these dichotomized neuralactivity networks does not allow for evaluation of strong and weakconnections relative to one another, and fails to account for negativeconnections (e.g., inhibitory connections). Additionally, the thresholdcan be subjective, with little or no theory informing the value of thethreshold. Furthermore, the use of different thresholds can lead toqualitatively different results.

Neural activity data can be evaluated according to graph theory. Graphtheory can be used to quantify structural and functional networks. Ingeneral, the brain or some subset thereof can be divided into a seriesof nodes (e.g., individual voxels, or clusters of voxels). Aconnectivity matrix can be created by evaluating a connection strengthbetween the series of nodes. The information encoded in the connectivitymatrix is thresholded with individual correlation values rounded to zeroor one to create an adjacency matrix of dichotomous connections, orcontinuous ties. A number of different statistics can be applied toconstructed networks to assess both network structure and the roles ofindividual nodes within the network. The metrics can be compared betweenindividuals, tasks, or both, to quantify network structural shifts.However, thresholding and a lack of statistical power, stand in the wayof fully utilizing the information provided by neural activity data.

SUMMARY

In an example, a computer-implemented method for modeling neuralarchitecture can include identifying, by one or more processors, regionsof interest of a brain of a subject based on image data characterizingthe brain of the subject. The computer-implemented method can furtherinclude mapping, by the one or more processors, the regions of interestto a connectivity matrix. The connectivity matrix can be a weighted anddirected or undirected network. The computer-implemented method canfurther include applying, by the one or more processors, a multivariatetransformation to the connectivity matrix to transform the connectivitymatrix into a partial correlation matrix. The multivariatetransformation can maintain a positive definite constraint for theconnectivity matrix. The computer-implemented method can further includetransforming, by the one or more processors, the partial correlationmatrix into a neural model indicative of the connectivity matrix.

In another example, a computer-implemented method can includeidentifying, by one or more processors, regions of interest of a brainof a subject based on image data. The image data can include one ofmagnetic resonance imaging (MRI) data, functional magnetic resonanceimaging (fMRI) data, diffusion tensor imaging (DTI) data, diffusionweighted imaging (DWI) data, electromyogram (EMG) data,electroencephalogram (EEG) data, positron emission tomography (PET)data, magnetoencephalography (MEG) data, Electrocorticography (ECoG)data, ultra-sound imaging data, and a combination thereof. The computerimplemented method can further include, mapping, by the one or moreprocessors, the regions of interest of the image data to a connectivitymatrix. The connectivity matrix can be a weighted and directed orundirected network. The computer-implemented method can further includetransforming, by the one or more processors, the connectivity matrixinto a biomarker parameter. The biomarker parameter can correspond to adefined statistic. The computer-implemented method can further includeanalyzing, by the one or more processors, the biomarker parameteraccording to a probability density function of a statistical cognitivemodel. The probability density function can map a separate population ofthe biomarker parameter to a cognitive phenomenon. Thecomputer-implemented method can further include generating, by the oneor more processors, a probability of the cognitive phenomenon based onthe probability density function of the statistical cognitive model. Theprobability can quantify a severity of the cognitive phenomenon.

In another example, a method can include receiving, at one or moreprocessors, image data of a subject. The image data can correspond toMRI image data. The method can further include, transforming, at the oneor more processors, the image data with a neural modeling engine into abiomarker parameter. The biomarker parameter can correspond to a definedstatistic. The method further includes analyzing, at the one or moreprocessors, the biomarker parameter with a probability density functionof a statistical cognitive model. The probability density function canmap a separate population of the biomarker parameter to a cognitivephenomenon. The method can further include generating, at the one ormore processors, an initial probability of the cognitive phenomenonbased on the probability density function of the statistical cognitivemodel. The method can further include quantifying, at the one or moreprocessors, a severity of the cognitive phenomenon for the subject basedupon the initial probability.

In an even further example, a method can include transforming, at one ormore processors, a connectivity matrix based on a statistic or set ofstatistics determined based on the connectivity matrix to a descriptivestatistical summary characterizing the connectivity matrix. The methodcan further include classifying, by the one or more processors, aseverity and type of cognitive phenomenon present in a subject based ona result of the comparison to a greater population of descriptivesummaries of connectivity matrices.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts an example of a system for modeling neural architecture.

FIG. 2 depicts an example of magnetic resonance imaging (MRI) data.

FIG. 3 depicts an example of a process for transforming a connectivitymatrix into a neural model.

FIG. 4 depicts an example of a flow diagram illustrating an examplemethod for generating a neural model and biomarker parameters.

FIGS. 5 and 6 depict examples of a process for mapping neural activitydata to a connectivity matrix.

FIG. 7 depicts an example of a flow diagram illustrating an examplemethod for generating a statistical cognitive model

FIG. 8 depicts an example of a flow diagram illustrating an examplemethod for analyzing a connectivity matrix.

DETAILED DESCRIPTION

Systems and methods are described herein for modeling neuralarchitecture.

FIG. 1 illustrates a system 10 for modeling neural architecture. Thesystem 10 can include an imaging device 20. The imaging device 20 can beconfigured to generate neural activity data 21 characterizing neuralconnectivity of a subject 24. The subject 24 can correspond to a human,an animal, or the like. In an example, the imaging device 20 can includea magnetic resonance imaging (MRI) scanner 22. The MRI scanner 22 can beconfigured to collect signals emitted from atomic nuclei of the subject24, and to generate images indicative of at least a portion of thesubject 24 based on the collected signals. As described herein, thegenerated images can correspond to the neural activity data 21. FIG. 1illustrates neural activity data 21 being provided based on imagesgenerated by the MRI scanner 22. However, it should be construed thatany neural activity technology (e.g., imaging technology) can be usedthat can produce either functional connectivity or structuralconnectivity data for the patient 24. Accordingly, any device that canprovide neural activity data 21 for the patient 24 based on capturedimages indicative of at least the portion of the subject 24 can be used.

The MRI scanner 22 can be configured to provide magnetic field signalswith a controlled field gradient, and to emit radio frequency (RF)signals that can interact with the atomic nuclei of the subject 24. TheMRI scanner 22 can include a magnet 26. The magnet 26 can be configuredto generate magnetic field signals that can be applied to the subject24. A magnetic moment of the atomic nuclei of the subject 24 cansubstantially align with a direction of the magnetic field based on themagnetic field signals. The MRI scanner 22 can further include gradientcoils 28. The gradient coils 28 can be configured to vary a magnitude ofthe magnetic field signals along one or more gradient directions withrespect to the subject 24. The atomic nuclei of the subject 24 caninclude a resonance frequency that can be proportional to the magnitudeof the magnetic field signals. The resonant frequency can depend on aposition along each of the one or more field gradients in response to anapplication of the one or more field gradients.

The MRI scanner 22 can further include RF coils 30. The RF coils 30 canbe configured to apply RF signals to the subject 24, and to receivesignals emitted by the subject 24. In an example, the RF signals can betransmitted at the resonant frequency of a spatial location along one ofthe field gradients. Thus, the atomic nuclei at the spatial location canchange a corresponding magnetization alignment relative to the magneticfield. Upon cessation of the RF signals by the RF coils 30, the magneticfield can urge the atomic nuclei back to a previous magnetizationalignment. As a result of varying the corresponding magnetizationalignment, the atomic nuclei of the subject 24 can emit signals (e.g., avarying magnetic flux). The RF coils can detect the emitted signals as achanging voltage.

The MRI scanner 22 can further include signal conditioning hardware 32.The signal conditioning hardware 32 can be coupled to the gradient coils28 and the RF coils 30. The signal conditioning hardware 32 can beconfigured to provide driving signals to the gradient coils 28 and theRF coils 30. Additionally, or alternatively, the signal conditioninghardware 32 can be configured to process the emitted signals detected bythe RF coils 30.

The MRI scanner 22 can further include one or more processors 34. Asused herein, the term “processor” can include any device that can beconfigured to executing machine readable instructions. The one or moreprocessors 34 can be configured to receive the emitted signals from theconditioning hardware 32. Thus, the one or more processors 34 can beconfigured to receive processed emitted signals. The one or moreprocessor 34 can further be configured to generate the neural activitydata 21 based on the emitted signals. In an example, the neural activitydata 21 can correspond to MRI image data 36, such as illustrated in FIG.2. In this example, the MRI image data 36 can be generated by the MRIscanner 22, as described herein. The MRI image data 36 can include MRIimages, functional magnetic resonance imaging (fMRI) images, diffusiontensor imaging (DTI) images, diffusion weighted images (DWI), acombination thereof, or the like.

The fMRI images can be used to providing mapping of active neuronswithin the brain of the subject 24. In particular, neural activityassociated with the brain of the subject 24 can be correlated with bloodflow and blood oxygenation in the brain. As neurons become active,oxygenated blood can flow into the neuron and displace deoxygenatedblood. Since oxygenated blood and deoxygenated blood can interactdifferently with magnetism, the signals emitted by the subject 24 can becorrelated to blood oxygenation, and thus, neural activity.Additionally, multiple fMRI images can be captured over time to mapneural response to stimulus.

In DTI and DWI images, a rate of diffusion of a moiety (e.g., water) canbe encoded in an intensity of each component of an image (e.g., voxel).During collection of emitted signals from the subject 24, a sufficientnumber of field gradients can be applied along gradient directions tocalculate a tensor, which can be indicative of the diffusion, for eachcomponent. The diffusion can be correlated to a structure of the brainof the subject 24, e.g., the structure can cause the diffusion to have adirectional preference. For instance, the structure of axons and myelinsheaths can be preferential to diffusion along a primary direction ofthe axons and myelin sheaths.

The data can further come from other brain imaging modalities such aselectroencephalogram (EEG) data, positron emission tomography (PET),magnetoencephalography (MEG), Electrocorticography (ECoG),electromyogram (EMG) data, and ultra-sound imaging. In EEG, electrodescan be placed on the subject's scalp to measure electrical potentialsgenerated by the brain. ECoG is similar, except the electrodes can beplaced directly on exposed brain (e.g., if the scalp has beentemporarily removed for a medical procedure). In MEG, magnetic fieldscan be recorded from the brain via magnetometers. In PET, radioactivemetabolic chemicals (e.g., glucose) can be injected into the subject'sbloodstream, and a corresponding emission from the radioactive metabolicchemicals can be subsequently measured to provide an indication of wherethe brain is using these chemicals.

The imaging device 20 can further include memory 38. The neutralactivity data 21 can be stored in the memory 38. The memory 38 can becoupled to the one or more processors 34. The one or more processors 34can be configured to retrieve the neutral activity data 21 stored in thememory 38. The memory 104 can be implemented, for example, as anon-transitory computer storage medium, such as volatile memory (e.g.,random access memory), non-volatile memory (e.g., a hard disk drive, asolid-state drive, flash memory or the like), or a combination thereof.

The imaging device 20 can further include network interface hardware 40.The network interface hardware 40 can be configured to couple theimaging device 20 to another device over a network 12. The network caninclude one of a wide area network (WAN), a local area network (LAN),personal area network (PAN), and combination thereof. The networkinterface hardware 40 can be configured to communicate, e.g., sendand/or receive data via a wired or wireless communication protocol basedon the network 12. For example, the network interface hardware 40 caninclude at least one of an antenna, a modem, LAN port, wireless fidelity(Wi-Fi) card, WiMax card, near-field communication hardware, and thelike, that can be configured to facilitate the sending and/or receivingof data to and from the imaging device 20. Thus, the imaging device 20can be coupled to the network 12 via physical wires, via a WAN, via aLAN, via a PAN, via a satellite network, or the like.

The system 10 can further include a neural architecture system 100. Theneural architecture system 100 can be configured to execute machinereadable instructions to provide inferential testing, network modeling,and/or network simulation of neural architecture based on the neuralactivity data 21. In an example, the neural architecture system 100 canbe implemented on a computer, a server, or the like. The neuralarchitecture system 100 can include one or more processors 102. The oneor more processors 102 can be coupled to memory 104 of the neuralarchitecture system 100. The one or more processors 102 can further becoupled to a network interface hardware 106 of the neural architecturesystem 100. In the present example, although the components of theneural architecture system 100 are illustrated as being implemented onthe same system, in other examples, the different components could bedistributed across different systems and communicate, for example, overthe network 12.

The neural architecture system 100 can be configured to be communicatewith one or more client devices 108 over the network 12. Each of the oneor more client devices 108 can include memory for storing data andmachine-readable instructions. The memory can be implemented similar tothe memory described herein. The one or more client devices 108 caninclude a tablet, a mobile device, a laptop, a computer, or the like.

The neural architecture system 100 can further include a neural modelingengine 110. The neural modeling engine 110 can be configured to generatea neural model 112 of the networks within the brain of the subject 24based on the neural activity data 21. The neural model 112 can includeany model that can simulate the neural activity data 21, e.g., theneural model 112 can provide an output that can resemble the neuralactivity data 21. The neural modeling engine 110 can include analgorithm. The algorithm can include, one of a deterministic algorithm,a probabilistic algorithm that does not account for networkinterdependencies, and probabilistic algorithm that does account fornetwork interdependencies. The algorithm can be applied to the neuralactivity data 21 to generate the neural model 112.

Many current approaches treat the neural activity data 21 as a matrix ofcorrelation values, which is necessarily undirected. While thedescriptive and inferential methods described herein are applicable tocorrelation networks, many of the descriptive and inferential examplesdescribed herein are similarly applicable to directed networks. Forexamples, techniques such as Granger Causality and Bayesian NetworkAnalysis can both produce estimates of the directed influence of onenode on another, as opposed to a single number reflecting the connectionbetween two nodes. The examples described herein are equally adaptableto either data format.

A node-level algorithm can specify a fixed degree distribution for thenodes of the brain and generate simulated networks based on the fixeddegree distribution. A probabilistic algorithm, which does not accountfor network interdependencies, can include linear regression, which caninclude exogenous or dyadic endogenous factors as inputs, and can usepredicted edge values to generate simulated networks. A probabilisticalgorithm, which can account for network interdependencies, can takeendogenous (e.g., network) and exogenous dependencies as inputs and cangenerate simulations of neural networks based on estimated parameterscorresponding to the inputs.

In some example, the neural modeling engine 110 can include aneurological generalized exponential random graph model (neuro-GERGM)algorithm. The neuro-GERGM algorithm can be classified as aprobabilistic algorithm, which can account for networkinterdependencies. The neuro-GERGM technique can create a jointprobability distribution by evaluating a set of interdependenciesbetween nodes of the brain, which can be specified as statistics on thenetwork of interest. The joint probability distribution can be used toevaluate the estimated parameter values of interest, and simulate neuralnetworks that can include a similar resemblance to the neural activitydata 21.

The neuro-GERGM algorithm can be considered as a distinctive member of asimilar class as exponential random graph models (ERGM) and generalizedexponential random graph models (GERGM). ERGMs can provide forinferential analysis of networks. ERGMs are generative models ofnetworks in which a user can define processes, both endogenous to thestructure of the network, and exogenous to the structure of the network,which can give rise to the network's structure of connectivity. Given aset of user-defined network statistics that can be specified to capturegenerative features of the network, the ERGM can fit parametersaccording to a maximum likelihood. The ERGM model can allow for theinferential testing of hypotheses about which exogenous predictors andendogenous structures generated the data. Inferential testing canprovide more analytic analysis than descriptive analyses.

For example, the ERGM model can characterize an observed network as afunction of triadic closure—the degree to which having strongconnections with two different regions makes those regions likely to bestrongly connected. The model can be configured to estimate parametersfor each defined statistic, resulting in a set of parameters that canmaximize the likelihood of observing the given network. The model canfurther be configured to generate simulated networks, for use inevaluating model fit or testing counterfactuals about alternativestructure of the networks or values of predictors.

An ERGM framework can be limited by the requirement of non-negativeinteger networks in which connections are limited to integer counts.Thus, ERGM frameworks can include similar drawbacks experienced by otherapproaches that can include thresholding the network (e.g., excludinginformation about negative ties or fractional weights). Second, ERGMnetworks can provide poorly fitted models due to a lack of statisticscurrently available for neuroscience. As described herein, a neuro-GERGMframework does not require any censoring or thresholding, and can beused to produce substantially well-fitted models to the neural activitydata 21 in contrast to the ERGM framework. Moreover, the neuro-GERGMframework can simulate networks according to model parameters thatmaintain the structure of the neural activity data 21.

In a related example, the neural activity data 21 can be summarized by astatistic that can be computed on the network and can be descriptiverather than inferential. These descriptive statistics can relate to alocation or prominence of a particular node of the network, to therelationship between two nodes, or to the connectivity, clustering,diameter, or other properties of the network. An example of a statisticcapturing the prominence of a node is Eigenvector centrality. For anetwork composed of a set of nodes/vertices V and a set of edge valuesconnecting them E, the matrix A can be defined such that A_(ij) capturesthe value of vertex i's connection to vertex j. The eigenvectorcentrality of vertex i can be defined as:

$\begin{matrix}{{c_{e_{i}}(v)} = {\alpha{\sum\limits_{{\{{u,v}\}} \in E}{c_{e_{i}}(u)}}}} & \left( {{Eqation}\mspace{14mu} 1} \right)\end{matrix}$where the vector

c_(e_(i)) = (c_(e_(i))(1), …  , c_(e_(i))(N_(v)))^(T)can be a solution to the eigenvalue problem

Ac_(e_(i)) = α⁻¹C_(e_(i)).

The optimal choice of α⁻¹ is a greatest eigenvalue of A and hence c_(e)_(i) is a corresponding eigenvector. The absolute values of theseentries can be reported, which can be automatically be on the [0,1]interval by the orthonormality of eigenvectors. An example of adescriptive statistic pertaining to the relationship between two nodesis the path length between said respective nodes. An alternative exampleis a measure of structural equivalence. An example of a descriptivestatistic on the network could be a related clustering coefficient(e.g., the census of closed triads on the network) or a related diameter(e.g., the longest shortest path in the network).

In the foregoing example, the statistics that can describe the neuralactivity data 21 can be used, absent the neural model 112, to classifycognitive phenomena and measure a corresponding severity. In thisexample, a given descriptive statistic or set of statistics computed forthe subject 24 can be compared by the one or several processors 102 to areference distribution of other individuals for that same statistic orset of statistics. Based on where the subject 24 falls relative to thereference distribution, a cognitive phenomenon can be indicated and thedistance or some other measure between the descriptive statistical valuefor the subject 24 and some measure of the center or central area of thereference distribution can be used to measure the severity of thecognitive phenomenon.

The GERGM is an exponential family of probability distributions that canbe used to characterize the process of graphs with continuous-valuededges. For example, y can be considered a vector of length m=20(20−1)/2(e.g., a number of undirected edges contained in the modeled network),or m=20(20−1) (e.g., a number of directed edges contained in the modelednetwork) where each entry of y is a real-valued edge weight thatdescribes one connection of the neural activity data 21. It is notedthat the vector y can be scaled to fit with the dimensions of the neuralactivity data 21. A vector x can have a similar length m, whose entriescan be bounded between 0 and 1, and can represent the relationalstructure of the neural activity data 21.

The specification of a GERGM for an observed graph y can proceed in twosteps. First, a vector of statistics h(x) of length p<m can be specifiedto describe the structural patterns of the graph x. The choice of theseendogenous (network-level) statistics can be flexible, and can bespecified to model any dyadic feature of a weighted graph. Once thestatistics have been specified, the bounded network x∈ [0, 1]m can bemodeled by a density function:

$\begin{matrix}{{{f_{X}\left( x \middle| \theta \right)} = \frac{\exp\left( {\theta^{T}{h(x)}} \right)}{\int_{{\lbrack{0,1}\rbrack}^{m}}{{\exp\left( {\theta^{T}{h(z)}} \right)}{dz}}}},} & \left( {{Equation}\mspace{14mu} 2} \right)\end{matrix}$where θ is a p-dimensional vector of parameters that quantifies theeffects of the network statistics h(x) on the likelihood of the graph x.

In the second specification step of the GERGM, a cumulative distributionfunction (CDF) T(·|β):

^(m)→[0, 1]^(m) can be selected to model a transformation from y to x.The transformation can be parameterized by a vector of q<m real-valuedparameters β. The function T(·|β) can be used to model the relationshipbetween the graph y on external, or exogenous, predictors that describethe network. For example, T can be selected as a joint cumulativedistribution function (CDF) of independent Gaussian random variables.Accordingly, a linear regression of y on any number of exogenouspredictors can be modeled and β can serve as the regression coefficientsassociated with the exogenous predictors.

Using the transformation function T, the edge weight between distinctnodes i and j can be modeled by the transformation x_(ij)=T_(ij) (y|β).By Equation 2, it can follow that the network y∈

^(m) can have a joint density function, such as

$\begin{matrix}{{{f_{Y}\left( {\left. y \middle| \theta \right.,\beta} \right)} = {{f_{X}\left( {T\left( y \middle| \beta \right)} \middle| \theta \right)}{\prod\limits_{ij}{t_{ij}\left( y \middle| \beta \right)}}}},} & \left( {{Equation}\mspace{14mu} 3} \right)\end{matrix}$where t_(ij)(y|β)=dT_(ij)(y|β)/dy_(ij) is a marginal density functionfor all i≠j. In the case that there are no structural dependencies(e.g., when θ=0), Equation 3 can be simplified to a regression of y onexogenous predictors.

FIG. 3 depicts an example of a process for transforming a connectivitymatrix into a neural model (e.g., the neural model 112, as shown in FIG.1). As shown in FIG. 3, the GERGM can transform the edges of aconnectivity matrix ρ of the neural activity data 21 onto atwo-dimensional space, such as a [0, 1] space, and calculate a partialcorrelation network ϕ based on the log-likelihood of the structuralparameters. Subsequently, the edges of the constrained network can betransformed back to the original space based on a regression on thespecified exogenous predictors.

Equation 3 can provide a framework to model the process of weightedgraphs. The GERGM framework can be modified to model correlationnetworks as the neuro-GERGM to analyze the neuronal architecture offunctional networks arising based on the neural activity data 21. Onedifficulty of modeling correlation matrices lies in the fact thatcorrelation matrices must be positive definite. The statistical modelsfor correlation networks should be constructed to account for theconstraint of positive definiteness. Unconstrained models offer limitedpredictive utility, and thus cannot be used to simulate validcorrelation matrices. The ERGM framework does not offer models that cangenerate networks constrained to be positive definite. Moreover,directly analyzing the structural properties of a brain correlationnetwork without accounting for positive definiteness can lead toinferential false positives due to artifacts in the data.

Suppose that ρ is the observed connectivity matrix of the neuralactivity data 21, ρ can be viewed as a weighted and undirected networkwhose edge weights can lie between −1 and 1. The neuro-GERGM algorithmcan evaluate the pairwise correlations among n regions in the brainarising based on the neural activity data 21. The neural activity data21 can be represented with an undirected network represented by theweighted symmetric adjacency matrix ρ=(ρ_(i,j)∈ 1, . . . , n). In thisexample, ρ_(ij)∈ [−1, 1] and ρ_(ij)=ρ_(ij) for all i and j. As describedherein, the evaluation of ρ applies the constraint of the correlationmatrices being positive definite. Thus, for any non-zero vector v oflength n, the following relation must hold:v ^(T) pv>0.  (Equation 4)

The relation can strongly constrain the values of ρ by includingstructural dependence between each entry ρ_(ij) and every other entry.Furthermore, for valid inference of ρ, the model should ensure thatestimates, for example {circumflex over (ρ)}, can satisfy Equation 4.

The neuro-GERGM framework can provide a family of models for a randommatrix ρ∈ [−1, 1]^(n×n) that can uphold the positive definite constraintof ρ, can characterize the relational (network) structure of ρ, and therelationship between the entries of ρ and a collection of potentiallyuseful regressors z₁, . . . , z_(q) For example, to satisfy the positivedefinite constraint on ρ, a multivariate transformation can be appliedto the observed network ρ to obtain the vector of unconstrained partialcorrelation values ϕ. The partial correlation network can be normalized,and directly modeled based on the GERGM specification in Equation 3. Theresulting graphical model can uphold the positive definite constraint onβ, and can describe both the endogenous and exogenous features of theobserved network.

In particular, ρ can be viewed as an undirected and weighted networkwhose edge weights (p_(ij):i<j∈ [n]) lie in [−1, 1]^(m), wherem=n(n−1)/2 is a total number of interactions among the n regions in thebrain. The neuro-GERGM framework, similar to the GERGM framework, canbegin with characterizing the joint network features of the connectivitymatrix ρ based on a constrained network x∈ [0, 1]^(m), which can have anexponential family density function consistent with Equation 2. A vectorof unconstrained partial correlations ϕ=: (ϕ_(ij):i<j∈[n])∈ [−1, 1]^(m)can be generated based on the transformationx_(ij)=T_(ij)((ϕ+1)/2|μ_(ij), α), where T_(ij) (·|μ_(ij), α) is the CDFof a Beta distribution with mean μ_(ij) and scale parameter α. Let T:

^(m)→[0, 1]^(m) can be defined as an m-dimensional vector T (ϕ|μ,α)=(T_(ij)(ϕ_(ij)|μ_(ij), α), i<j∈[n]). A random vector form can begenerated based on the probability density function (PDF) f₉₉ (ϕ|θ, μ,α):

$\begin{matrix}{{{{{f_{\phi}\left( {\left. \phi \middle| \theta \right.,\mu,\alpha} \right)} = {\frac{\exp\left( {\theta^{T}{h\left( {T\left( {\left. {\left( {\phi + 1} \right)/2} \middle| \mu \right.,\alpha} \right)} \right)}} \right)}{2\;{\int_{{\lbrack{0,1}\rbrack}^{m}}{{\exp\left( {\theta^{T}{h(z)}} \right)}{dz}}}}{\prod\limits_{ij}{t_{ij}\left( {\left. {\left( {\phi + 1} \right)/2} \middle| \mu_{ij} \right.,\alpha} \right)}}}},{{\phi \in \left\lbrack {{- 1},1} \right\rbrack^{m}};}}{t_{ij}\left( {\left. w \middle| \mu_{ij} \right.,\alpha} \right)} = {\frac{\left\lceil (\alpha) \right.}{\left\lceil {\left( {\mu_{ij}\alpha} \right)\left\lceil \left( {\left( {1 - \mu_{ij}} \right)\alpha} \right) \right.} \right.}{w_{ij}^{{\mu_{ij}\alpha} - 1}\left( {1 - w_{ij}} \right)}^{{{({1 - \mu_{ij}})}\alpha} - 1}}},\mspace{20mu}{w_{ij} \in \left( {0,1} \right)}} & \left( {{Equation}\mspace{14mu} 5} \right)\end{matrix}$where Γ(t)=f₀ ^(∞)x^(t−t)e^(−x)dx is a standard gamma function.

Equation 5 can be related to the GERGM formulation of Equation 3, withan exception that T is selected as a joint Beta CDF to account for thedomain of the partial correlation values in ϕ. A notable property ofEquation 5 is that when the partial correlation network ϕ does notcontain any network structure, e.g., when θ=0, then the m components of(ϕ+1)/2 are independent samples from a Beta(μ_(ij), α) distribution,where μ_(ij)∈ (0, 1) is the expected value of (ϕ_(ij)+1)/2 and α is theprecision of the distribution.

Equation 5 can be used to evaluate the effect of exogenous variables z₁,. . . , z_(q) on the partial correlation matrix ϕ. Let z₁, be a vectorof dyadic observations z_(l), be a vector of dyadic observationsz₁=(z_(lj)(I):i<j∈[n]) for l=1, . . . , q. The effect of each predictorcan be modeled based on a linear model for the mean μ of ϕ:log it(μ_(ij))=β₀+β₁ z _(ij)(1)+ . . . +β_(q) z _(ij)(q)  (Equation 6)

As μ_(ij)∈ (0, 1), the log it (·) link in Equation 5 is an option, butany monotonically increasing and twice differentiable function can beused. The regression model in Equation 6, together with the Beta densityt_(ij)(·|μ_(ij), α) can be generally referred to as “Beta regression.”

The final step of defining the neuro-GERGM framework is based on arelationship between the partial correlation matrix ϕ and an associatedconnectivity matrix ρ for the collection of n brain regions. Forinstance, ϕ_(j j+k) can denote the partial correlation between region jand j+k given regions j+1, . . . j+k−1 for k≥2. One-to-one mapping canbe used between ρ and ϕ. For example, given ϕ, the connectivity matrix ρcan be computed based on the following recursion:

$\begin{matrix}{\rho_{{j\mspace{14mu} j} + k} = \left\{ {\begin{matrix}1 & {k = 0} \\\phi_{{j\mspace{14mu} j} + k} & {k = 1} \\{{{\rho_{1}^{T}\left( {j,k} \right)}D_{jk}^{- 1}{\rho_{2}^{T}\left( {j,k} \right)}} + {\phi_{{j\mspace{14mu} j} + k}C_{jk}}} & {2 \leq k \leq {n - j}}\end{matrix},\mspace{20mu}{{{where}\mspace{20mu}{\rho_{1}^{T}\left( {j,k} \right)}} = \left( {\rho_{{j\mspace{14mu} j} + 1},\ldots\mspace{14mu},\rho_{{j\mspace{14mu} j} + k - 1}} \right)},\mspace{20mu}{{\rho_{2}^{T}\left( {j,k} \right)} = \left( {\rho_{j + {k\mspace{14mu} j} + 1},\ldots\mspace{14mu},\rho_{j + {k\mspace{14mu} j} + k - 1}} \right)},{C_{jk}^{2} = {\left\lbrack {1 - {{\rho_{1}^{T}\left( {j,k} \right)}D_{jk}^{- 1}{\rho_{1}^{T}\left( {j,k} \right)}}} \right\rbrack{\quad{\left\lbrack {1 - {{\rho_{2}^{T}\left( {j,k} \right)}D_{jk}^{- 1}{\rho_{2}^{T}\left( {j,k} \right)}}} \right\rbrack,\mspace{20mu}{{{and}\mspace{20mu} D_{jk}} = \begin{pmatrix}p_{j + {1\mspace{14mu} j} + 1} & p_{j + {2\mspace{14mu} j} + 1} & \ldots & p_{j + k - {1\mspace{14mu} j} + 1} \\p_{j + {1\mspace{14mu} j} + 2} & p_{j + {2\mspace{14mu} j} + 2} & \ldots & p_{j + k - {1\mspace{14mu} j} + 2} \\\vdots & \vdots & \vdots & \vdots \\p_{j + {1\mspace{14mu} j} + k - 1} & p_{j + {2\mspace{14mu} j} + k - 1} & \ldots & p_{j + k - {1\mspace{14mu} j} + k - 1}\end{pmatrix}}}}}}} \right.} & \left( {{Equation}\mspace{14mu} 7} \right)\end{matrix}$The multivariate transformation from ρ to ϕ given based on the recursionin Equation 7 can be represented by ρ=g(ϕ). Applying the inverseprobability transform to the density in Equation 5, a positive definiteconnectivity matrix ρ can be generated based on the density:

$\begin{matrix}{D_{jk} = \begin{pmatrix}p_{j + {1\mspace{14mu} j} + 1} & p_{j + {2\mspace{14mu} j} + 1} & \ldots & p_{j + k - {1\mspace{14mu} j} + 1} \\p_{j + {1\mspace{14mu} j} + 2} & p_{j + {2\mspace{14mu} j} + 2} & \ldots & p_{j + k - {1\mspace{14mu} j} + 2} \\\vdots & \vdots & \vdots & \vdots \\p_{j + {1\mspace{14mu} j} + k - 1} & p_{j + {2\mspace{14mu} j} + k - 1} & \ldots & p_{j + k - {1\mspace{14mu} j} + k - 1}\end{pmatrix}} & \left( {{Equation}\mspace{14mu} 8} \right)\end{matrix}$where |J(c)| is the determinant of the Jacobian of the transformationgiven by:

$\begin{matrix}{{{J(c)}} = \left\lbrack {\prod\limits_{i = 1}^{n - 1}\;{\left( {1 - c_{{ii} + 1}^{2}} \right)^{n - 1}{\prod\limits_{k = 2}^{n - 2}\;{\prod\limits_{i = 1}^{n - k}\;\left( {1 - c_{{ii} + k}^{2}} \right)^{n - k - 1}}}}} \right\rbrack^{{- 1}/2}} & \left( {{Equation}\mspace{14mu} 9} \right)\end{matrix}$

Accordingly, for fixed parameters θ=(θ₁, . . . , θ_(ρ)), β=(β₀, . . . ,β_(q)), and α>0, the neuro-GERGM model for a connectivity matrix ρ on nbrain regions can be characterized by the following process:ρ˜f _(R)(r|θ,β,α)log it(μ_(ij))=β₀+β₁ z _(ij)(1)+ . . . +β_(q) z_(ij)(q),i<j∈[n]  (Equation 10)

To fit a neuro-GERGM model in Equation 10 to an observed connectivitymatrix ρ, statistics can be defined, computed either on the network ofinterest or between the network of interest, and another set of values.In an example, the statistics can include two forms: endogenous(network-level) statistics h(x)=(h₁(x), . . . , h_(ρ)(x)), and exogenousstatistics z₁, . . . , z_(q). Accordingly, to fit the neuro-GERGM model,the endogenous predictors can be specified as the vector h(x), theexogenous predictors can be specified for the regression functionT(·|β), or both. The corresponding vectors of coefficients, θ and β, cancharacterize the effect of the specified statistics on the process ofthe observed weighted graph.

Endogenous statistics can include statistics for which the derivative ofthe statistic value with respect to the value of at least one edge inthe network depends on the value of at least one other edge in thenetwork. The endogenous statistics can be computed as sums of subgraphproducts such as, for example, two-stars, transitive triads, or anyother statistic suitable to characterize neural activity. The two-starsstatistic can measure preferential attachment, which can be higher fornetworks in which there is a high variance in the popularity ofvertices. Two-stars is relevant to neuroscience research in graphs sincetwo-stars can correspond to the degree to which hubs are present in agiven network relative to chance, allowing for direct inferentialtesting of the role of hubs in a given network. Two-stars can quantifythe presence of hubs in a network given the importance for hubs inneural networks. Transitive triads (triads) can measure clustering,which can be higher when triads of nodes exhibit higher interconnectionbetween them. As with preferential attachment, clustering has been shownto be highly important component of neural networks, with the brainshowing generally a high-degree of clustering. Further, reductions inclustering (in addition to other metrics), have been associated withdisease states such as, for example, schizophrenia.

Exogenous statistics can include statistics for which the derivative ofthe statistic with respect to the value of any edge in the network doesnot depend on the value of any other edge in the network. Endogenousstatistics can include covariates that can allow for specification ofspecific node, dyad, or network level properties. Specifying covariatescan yield a parameter estimate that corresponds to the magnitude of thatcovariate on strength of edges. At the node level, either categorical orcontinuous properties can be specified for each node, and then acovariate can be added based on the relationship between node values.For example, a user can define whether nodes are in the left hemisphere,the right hemisphere, or on the midline. The impact of hemisphere usecan be modeled as a node-match term, which can code whether two nodesshare a property or not. Thus, nodes can be evaluated to determinewhether the nodes have stronger connections to nodes in the samehemisphere rather than the opposing hemisphere. At the network level,the nodes located near each other in a physical space can be modeled todetermine whether the nodes are more connected than those that are not.For example, a network level covariate in a matrix of the distancesbetween each node. Additionally, or alternatively, the distances can becalculated using Euclidean distance relative to a center of each regionof interest (ROI).

Adding in exogenous statistics to the neuro-GERGM framework can provideseveral advantages. Adding in exogenous covariates can improve thelikelihood of model convergence. Additionally, without exogenouscovariates, the simulated networks can have similar structure to theobserved network, but will not recover specific edges (e.g., nodes willbe exchangeable). Moreover, exogenous covariates allow theoreticaltesting of whether properties of nodes or dyads influence the strengthof connectivity between those regions. By including an exogenouscovariate, inferential testing of the importance of such a covariate canbe provided. Taken together, by combining endogenous and exogenousstatistics, the neuro-GERGM framework can be used to model a wide rangeof network interdependencies in weighted networks.

Referring again to FIGS. 1 and 3, the neural modeling engine 110 can beprogrammed to transform a connectivity matrix ρ corresponding to theneural activity data 21 into the neural model 112 according to theneuro-GERGM. The connectivity matrix ρ, and the statistics can beprovided to the neural modeling engine 110. Estimation of theneuro-GERGM can include the use of approximate computational methods asa result of the computationally intractable normalizing constant in adenominator of Equation 2. In some examples, the Metropolis-HastingsMarkov Chain Monte Carlo methodology can be used to calculate themaximum likelihood estimators for the fixed unknown parameters θ and β.The resulting parameter estimates {circumflex over (θ)} and {circumflexover (β)} can characterize the effects of the endogenous and exogenousstatistics on the likelihood of ρ, respectively. These parameterestimates can be used to further explore the nature of the data throughsimulation and testing.

It is noted that, while an exemplary computational method is describedherein with respect to the GERGM, the computational method can also beused for the neuro-GERGM. Given a specification of statistics h(·),transformation function T⁻¹, and observations Y=y from Equation 3, themaximum likelihood estimates (MLEs) of the unknown parameters θ and βcan be determined, namely values {circumflex over (θ)} and {circumflexover (β)} that maximize the log likelihood according to:

$\begin{matrix}{{{{\ell\left( {\theta,\left. \beta \right|} \right)} = {{\theta^{\prime}{h\left( {T\left( {,\beta} \right)} \right)}} - {\log\mspace{14mu}{C(\theta)}} + {\sum\limits_{ij}{\log\mspace{14mu}{t_{ij}\left( {,\beta} \right)}}}}},\mspace{20mu}{where}}\mspace{20mu}{{C(\theta)} = {\int_{{\lbrack{0,1}\rbrack}^{m}}{{\exp\left( {\theta^{\prime}{h(z)}} \right)}{{dz}.}}}}} & \left( {{Equation}\mspace{14mu} 10} \right)\end{matrix}$

The maximization of Equation 10 can be achieved through alternatemaximization of β|θ and θ|β. For example, the MLEs {circumflex over (θ)}and {circumflex over (β)} can be computed by iterating between thefollowing two steps until convergence. For r≥1, iterate untilconvergence:

1. Given θ^((r)), estimate β^((r))|θ^((r)):

$\begin{matrix}{\beta^{(r)} = {\arg\;{\max_{\beta}{\left( {{\theta^{(r)}{h\left( {T\left( {,\beta} \right)} \right)}} + {\sum\limits_{ij}{\log\mspace{14mu}{t_{ij}\left( {,\beta} \right)}}}} \right).}}}} & \left( {{Equation}\mspace{14mu} 10} \right)\end{matrix}$2. Set {circumflex over (x)}=T(y,β^((r))). Then estimateθ^((r+1))|β^((r)):θ^(r+1)=argmax_(θ)(θ′h({circumflex over (x)})−log C(θ)).  (Equation 12)

For fixed θ, the likelihood maximization in Equation 5 can beaccomplished numerically using gradient descent. In an example, whereint_(ij) is log-concave, and h ° T is concave in β, a hill climbingalgorithm can be used to find the global optimum. To estimate themaximization in Equation 12, which can be a difficult problem to solvedue to the intractability of the normalization factor C(θ) the MarkovChain Monte Carlo (MCMC) can be applied. As described herein, θ can beestimated according to the MCMC framework.

Let θ and {tilde over (θ)} be two arbitrary vectors in

^(p) and let C(·) be defined as in Equation 10. The crux of optimizingEquation 12 based on Monte Carlo simulation relies on the followingproperty of exponential families:

$\begin{matrix}{\frac{C(\theta)}{C\left( \overset{\sim}{\theta} \right)} = {{{\mathbb{E}}_{\overset{\sim}{\theta}}\left\lbrack {\exp\left( {\left( {\theta - \overset{\sim}{\theta}} \right)^{\prime}{h(X)}} \right)} \right\rbrack}.}} & \left( {{Equation}\mspace{14mu} 13} \right)\end{matrix}$

The expectation in Equation 13 cannot be directly computed. However, afirst order approximation for this quantity can be given by the firstmoment estimate:

$\begin{matrix}{{{{\mathbb{E}}_{\overset{\sim}{\theta}}\left\lbrack {\exp\left( {\left( {\theta - \overset{\sim}{\theta}} \right)^{\prime}{h(X)}} \right)} \right\rbrack} \approx {\frac{1}{M}{\sum\limits_{j = 1}^{M}{\exp\left( {\left( {\theta - \overset{\sim}{\theta}} \right)^{\prime}{h\left( x^{(j)} \right)}} \right)}}}},} & \left( {{Equation}\mspace{14mu} 14} \right)\end{matrix}$where x⁽¹⁾, . . . , x^((M)) is an observed sample from pdf f_(x) (·,{tilde over (θ)}).

Define

(θ|{circumflex over (x)}):=θh({circumflex over (x)})−log C (θ). Thenmaximizing

(θ|{tilde over (x)}) with respect to θ∈

^(p) R is equivalent to maximizing

(θ|{circumflex over (x)})−

({tilde over (θ)}|{circumflex over (x)}) for any fixed arbitrary vector{tilde over (θ)}∈

^(p) p. Equations 13 and 14 suggest:

$\begin{matrix}{{{\ell\left( \theta \middle| \overset{\sim}{x} \right)} - {\ell\left( \overset{\sim}{\theta} \middle| \hat{x} \right)}} \approx {{\left( {\theta - \overset{\sim}{\theta}} \right)^{\prime}{h\left( \hat{x} \right)}} - {{\log\left( {\frac{1}{M}{\sum\limits_{j = 1}^{M}{\exp\left( {\left( {\theta - \overset{\sim}{\theta}} \right)^{\prime}{h\left( x^{(j)} \right)}} \right)}}} \right)}.}}} & \left( {{Equation}\mspace{14mu} 15} \right)\end{matrix}$

An estimate for θ can be calculated by the maximization of (8). Ther+1st iteration estimate θ^((r+1)) in 5 can be computed based on theMonte Carlo methods. For example, r+1st iteration estimate θ^((r+1)) in5 can be computed by iterating between the following two steps: Givenβ^((r)), θ^((r)), and {circumflex over (x)}=T(y, β^((r))).

1. Simulate networks x⁽¹⁾, . . . , x^((M)) from density fx(x, θ^((r))).

2. Update:

$\begin{matrix}{\theta^{({r + 1})} = {\arg\;{\max_{\theta}{\left( {{\theta^{\prime}{h\left( \hat{x} \right)}} - {\log\left( {\frac{1}{M}{\sum\limits_{j = 1}^{M}{\exp\left( {\left( {\theta - \theta^{(r)}} \right)^{\prime}{h\left( x^{(j)} \right)}} \right)}}} \right)}} \right).}}}} & \left( {{Equation}\mspace{14mu} 16} \right)\end{matrix}$

Given observations Y=y, the Monte Carlo algorithm described above canrequire an initial estimate β⁽⁰⁾ and β⁽¹⁾. β₍₀₎ can be initialized usingEquation 11 in the case that there are no network dependencies present,namely, β⁽⁰⁾=argmax_(β) {Σ_(ij) log t_(ij)(y,β)}. x_(obs)=T(y,β₀) can befixed, and the Robbins-Monro algorithm for exponential random graphmodels can be used to initialize θ⁽¹⁾. This initialization step can bethought of as the first step of a Newton-Raphson update of the maximumpseudo-likelihood estimate (MPLE) θ_(MPLE) on a small sample of networksgenerated from the density f_(x) (x_(obs), θ_(MPLE)).

The Monte Carlo algorithm can include a simulation based on the densityf_(x)(x, θ^((r))). As this density cannot be directly computed, MCMCmethods such as, for example, Gibbs or Metropolis-Hastings samplers, canbe used to simulate from f_(x)(x, θ^((r))). The Gibbs sampling procedurecan be used in estimating θ by providing the sample of M networks to beused in approximating Equation 15. However, Gibbs sampling procedure canrestrict the specification of network statistics h(·) in the GERGMformulation. For example, the use of Gibbs sampling requires that thenetwork dependencies in an observed network y are captured through x byfirst order network statistics, namely statistics h(·) that are linearin X_(ij) for all i, j∈ [n]. A closed-form conditional distribution ofX_(ij) given the remaining network, X-_((ij)) can be computed accordingthis assumption which can be used in Gibbs sampling.

For example, fx_(ij)|x-_((ij)) (x_(ij), θ) can denote the conditionalpdf of X_(ij) given the remaining restricted network X-(_(ij)). Considerthe following condition on h(x):

$\begin{matrix}\begin{matrix}{{\frac{\partial^{2}{h(x)}}{\partial x_{ij}^{2}} = 0},} & {i,{j \in \lbrack n\rbrack}}\end{matrix} & \left( {{Equation}\mspace{14mu} 17} \right)\end{matrix}$

Assuming that Equation 17 holds (e.g., is true), a closed formexpression can be derived for fx_(ij)|x-_((ij))(x_(ij), θ):

$\begin{matrix}{{f_{X_{ij}|X_{- {({ij})}}}\left( {x_{ij},\theta} \right)} = \frac{\exp\left( {x_{ij}\frac{\theta^{\prime}{\partial{h(x)}}}{\partial x_{ij}}} \right)}{\left( {\theta^{\prime}\frac{\partial{h(x)}}{\partial x_{ij}}} \right)^{- 1}\left\lbrack {{\exp\left( {\theta^{\prime}\frac{\partial{h(x)}}{\partial x_{ij}}} \right)} - 1} \right\rbrack}} & \left( {{Equation}\mspace{14mu} 18} \right)\end{matrix}$

Let U be uniform on (0, 1). According to the conditional density in(11), values of x∈

^(m) can be simulated iteratively by drawing edge realizations ofX_(ij)|X-_((ij)), based on the following distribution:

$\begin{matrix}{\left. X_{ij} \middle| {X_{- {ij}} \sim \frac{\log\left\lbrack {1 + {U\left( {{\exp\left( {\theta^{\prime}\frac{\partial{h(x)}}{\partial x_{ij}}} \right)} - 1} \right)}} \right\rbrack}{\theta^{\prime}\frac{\partial{h(x)}}{\partial x_{ij}}}} \right.,{{\theta^{\prime}\frac{\partial{h(x)}}{\partial x_{ij}}} \neq 0}} & \left( {{Equation}\mspace{14mu} 19} \right)\end{matrix}$When

${{\theta^{\prime}\frac{\partial{h(x)}}{\partial h_{ij}}} = 0},$all values in [0,1] are equally likely; thus, X_(ij)|X-_((ij)) can bedrawn uniformly from support [0,1]. The Gibbs simulation procedure cansimulate network samples x⁽¹⁾, . . . , x^((M)) from fx(x, θ) bysequentially sampling each edge from its conditional distributionaccording to Equation 19.

Equation 17 can restrict the class of models that can be fit under theGERGM framework. To appropriately fit structural features of a networksuch as the degree distribution, reciprocity, clustering or assortativemixing, it may be necessary to use network statistics that involvenonlinear functions of the edges. Under Equation 16, nonlinear functionsof edges are not permitted—a limitation that may prevent theoreticallyor empirically appropriate models of networks in some domains. To extendthe class of available GERGMs, Metropolis-Hastings can be used.

An alternative and more flexible way to sample a collection of networksbased on the density fx(x, θ) is the Metropolis-Hastings procedure. TheMetropolis-Hastings procedure can include sampling the t+1st network,x^((t+1)), via a truncated multivariate Gaussian proposal distributionq(·|x^((t)) whose mean depends on the previous sample x^((t)) and whosevariance is a fixed constant σ².

An advantage of the truncated Gaussian for bounded random variables overthe Beta distribution is that it is straightforward to concentrate thedensity of the truncated Gaussian around any point within the boundedrange. For example, a truncated Gaussian with μ=0.75 and σ=0.05 canresult in proposals that are nearly symmetric around 0.75 and staywithin 0.6 and 0.9. The shape of the Beta distribution can be lessamenable to precise concentration around points within the unitinterval, which can lead to problematic acceptance rates in theMetropolis-Hastings algorithm.

Sample w can be based on a truncated normal distribution on [a, b] withmean μ and variance σ² (written W˜TN_((a,b))(μ, σ²)) if the PDF of W isgiven by:

$\begin{matrix}{{{g_{W}\left( {\left. \omega \middle| \mu \right.,\sigma^{2},a,b} \right)} = \frac{\sigma^{- 1}{\phi\left( \frac{\omega - \mu}{\sigma} \right)}}{{\Phi\left( \frac{b - \mu}{\sigma} \right)} - {\Phi\left( \frac{a - \mu}{\sigma} \right)}}},{a \leqslant \omega \leqslant b}} & \left( {{Equation}\mspace{14mu} 20} \right)\end{matrix}$where ϕ(·|μ, σ²) is the pdf of a N(μ, σ²) random variable and Φ(·) isthe cdf of the standard normal random variable. The truncated normaldensity on the unit interval can be represented asq _(σ)(ω|x)=g _(w)(ω|x,σ ²,0,1)  (Equation 21)

Equation 21 can correspond to the proposal density. Denote the weightbetween node i and j for sample t by x_(ij) ^((t)). TheMetropolis-Hastings procedure can further include generating samplex^((t+1)) sequentially according to an acceptance and/or rejectionalgorithm. The t+1st sample x^((t+1)) can be generated as follows:

1. For i, j∈ [n], generate proposal edge {tilde over (x)}_(ij) ^((t)˜q)_(σ)(w|x_(ij) ^((t))) independently across edges.

2. Set

x ( t + 1 ) = { x ~ ( t ) = ( x ~ ij ( t ) ) i , j ∈ [ n ] w . p . ⁢ ρ ⁡ (x ( t ) , x ~ ( t ) ) x ( t ) w . p . ⁢ 1 - ρ ⁡ ( x ( t ) , x ~ ( t ) ) ⁢ ⁢ρ ⁡ ( x , y ) = ⁢ min ( f X ⁡ ( | θ ) f X ⁡ ( | θ ) ⁢ ∏ i , j ∈ [ n ] ⁢ ⁢ q σ ⁡( x ij | ij ) q σ ⁡ ( ij | x ij ) , 1 ) = ⁢ min ( exp ⁡ ( θ ′ ⁡ ( h ⁡ ( ) - h⁡( x ) ) ) ⁢ ∏ i , j ∈ [ n ] ⁢ ⁢ q σ ⁡ ( x ij | ij ) q σ ⁡ ( ij | x ij ) , 1 )( Equation ⁢ ⁢ 22 )

The acceptance probability p(x^((t)), {tilde over (x)}^((t))) can bethought of as a likelihood ratio of the proposed network given thecurrent network x^((t)) and the current network given the proposal{tilde over (x)}^((t)). Large values of p(x^((t)), {tilde over(x)}^((t))) suggest a higher likelihood of the proposal network. Theresulting samples {x^((t)), t=1, . . . , M} define a Markov Chain whosestationary distribution is the target pdf fx(·|θ).

The proposal variance parameter σ² can influence the average acceptancerate of the Metropolis-Hastings procedure as described above. Forexample, the value of σ² can be inversely related to the averageacceptance rate of the algorithm. In some examples, the proposalvariance parameter σ² can be tuned such that the average acceptance rateis a given value, for example, approximately 0.25.

In view of the foregoing structural and functional features describedabove, a method that can be implemented will be better appreciated withreference to FIGS. 4, 7, and 8. While, for purposes of simplicity ofexplanation, the method of FIGS. 4, 7 and 8 is shown and described asexecuting serially, it is to be understood and appreciated that suchmethod is not limited by the illustrated order, as some aspects could,in other examples, occur in different orders and/or concurrently withother aspects from that shown and described herein. Moreover, not allillustrated features may be required to implement a method. The methodor portions thereof can be implemented as instructions stored in one ormore non-transitory storage media as well as be executed by a processingresource of a computer system, for example (e.g., one or more processorcores, such as the one or more processors 104 of the neural architecturesystem 100, as illustrated in FIG. 1, or the one or more processors of arelated device 108, as illustrated in FIG. 1).

FIG. 4 depicts an example of a method 200 for transforming aconnectivity matrix into a neural model. The method 200 can generate aneural model 112 and biomarker parameters 120 based on the neuralactivity data 21, such as illustrated in FIG. 1. At, 202, the neuralactivity data 21 can be generated. For example, the brain of the subject24 can be scanned by the imaging device 20, such as shown in FIG. 1. Inan example, the MRI scanner 22 can be configured to generate the MRIimage data 36, such as illustrated in FIG. 2. In an alternative example,any device capable of generating neural activity of the subject 24 canbe used. Thus, the neural activity data 21 can be generated by anycurrently existing (or future) device configured to generate neuralactivity data 21 based on the subject 24.

At 204, the neural activity data 21 can be preprocessed. For example,the neural activity data 21 can be preprocessed to remove noise andother undesired artifacts from the data collection. In some examples,preprocessing can be performed by the one or more processors 34 of theMRI scanner 22. Preprocessing can include, but not limited to, slicetiming correction, removal of motion artifacts, noise reduction,extraction, segmentation, and smoothing, or the like. In an example, thefMRI and DTI images can be registered and scaled to an MRI imagecapturing the structure of the subject 24 or a structural model.

At 206, one or more region of interest (ROI) in the neural activity data21 can be identified. The ROIs can correspond to any portion of theneural activity data 21 of known neurological importance such as, forexample, based upon the activity of the subject 24 according to a task,or identified in an atlas derived for use in neuroscience. The atlas canbe configured to integrate the results across different subject 24. Forexample, a common brain atlas can be used to adjust the neural activitydata 21 to the atlas for comparative analysis. In some examples, theatlas can be defined according to a network approach. The networkapproach can delineate and investigate critical subnetworks of the brainsuch as, for example, the Default Mode Network (DMN), the DorsalAttention Network, and the Frontoparietal Control Network. Networkanalysis can be used to investigate both the global and local structureof the brain, and the interaction of the networks with one another forindividuals both at rest and during tasks.

Network investigations of neural connectivity can reveal organizingprinciples of the whole brain. First, the brain can be considered to behighly modular, e.g., the network can be well separated into denselyconnected subnetworks, as discussed above. Furthermore, brain networkscan be hierarchically modular, e.g., subnetworks can be modular.Moreover, brain networks can demonstrate a “small world” structure: bothstrong clustering of regions within communities, and integration acrossthese communities. This feature of brain networks can be considered asbalancing local and global efficiency, and allowing the brain tominimize wiring costs. The brain can include a prevalence of “hub”regions: regions that are more highly interconnected with a large numberof other regions than expected by chance. For example, brains canexhibit a “rich club” property of these hubs, where some hubs are highlyinterconnected with one another.

The ROIs can be used to analyze cognitive phenomenon that can includecognitive processes, cognitive deficits, and cognitive disorders. Thenetwork approach can be used to analyze cognitive processes such as, forexample, learning, memory, cognitive control, emotion, and the like.Additionally, the network approach can identify network differences forindividuals suffering from cognitive disorders such as, for example,schizophrenia, concussion, traumatic brain injury, multiple sclerosis,and so on.

The network approach to neuroscience can be broadly categorizedaccording to the type of analysis at hand, including the analysis of (i)functional connectivity (e.g., correlations in signal across differentregions), or (ii) structural connectivity (e.g., investigation of whitematter tracks via DTI). It is noted that functional and structuralanalyses can be performed together. Functional connectivity can befurther divided into investigations of participants at “rest”—performingno task aside from staying awake and referred to as resting statefunctional connectivity—and investigations of network activity whileparticipants perform tasks. Task related investigations can detectmodulations of network activity as a function of task, and individualdifferences. Moreover, the network activity during a task can bepredictive of task performance.

Methods for quantifying network activity can include PsychophysiologicalInteraction (PPI), which can use functional connectivity between ROIs inorder to examine the interplay of regions. The functional connectivitycan be between a relatively small number of regions, or between a seedregion and the entire rest of the brain. Other methods for investigatingnetwork structure, that do not require the specification of a seedregion, an include independent components analyses (ICA) of time-seriesdata to identify networks of regions that co-activate across afunctional run. ICA methods can recover functional networks in adata-driven manner without using a predefined division of regions. Thisin turn allows for the probing of full functional networks rather thanhow a single region shifts connectivity with other regions.

At 208, the connectivity matrix ρ can be constructed based on the neuralactivity data 21. In some examples, the neural activity data 21 can bemapped to the connectivity matrix ρ according to the atlas used fordetermining the ROIs as a weighted network. For example, as illustratedin FIG. 5, when the neural activity data 21 includes an fMRI image, theconnectivity matrix ρ can be provided as a correlation matrix.Alternatively, or additionally, as illustrated in FIG. 6, when theneural activity data 21 includes a DTI image, the connectivity matrix ρcan be provided as the number of fiber tracks connecting the regions. Insome examples, the weights of connectivity matrix ρ can be scaled priorto being provided to the neural modeling engine 110, e.g., the weightscan be scaled to lie between −1 and 1.

At 210, the connectivity matrix ρ can be analyzed, for example, by theneural modeling engine 110. As described herein, the neural modelingengine 110 can be defined according to endogenous statistics andexogenous statistics. Accordingly, the neural modeling engine 110 can beprogrammed to transform the connectivity matrix ρ into the neural model112 representative of the neural activity data 21. Additionally, theneural modeling engine 110 can be programmed to transform theconnectivity matrix ρ into biomarker parameters 120 corresponding to theendogenous statistics and exogenous statistics and representative of theneural activity data 21. In some examples, a joint modeling frameworkcan be providing by combining connectivity matrices derived based ongiven ROIs (e.g., multiple distinct atlases). For example, a combinedconnectivity matrix can be created by combining the connectivity matrixρ derived based on an MRI image with the connectivity matrix ρ derivedbased on an fMRI image.

FIG. 7 depicts an example of a method for generating a statisticalcognitive model 122 of FIG. 1. The statistical cognitive model 122 canbe used to predict cognitive phenomenon such as, for example, cognitivedeficits, cognitive disorders, or both using the biomarker parameters120. The method 220 can begin at 222 by collecting population data fromone or more subjects 24. For example, instances of the neural activitydata 21, the biomarker parameters 120, and cognitive phenomena data 124can be collected to define a statistically significant population. Thecognitive phenomena data 124 can include diagnosis of a particularcognitive phenomena that can correspond to the neural activity data 21.The neural activity data 21, the biomarker parameters 120, and thecognitive phenomena data 124 can be associated with one another suchthat each instance of data can be identified based upon anotherinstance.

At 224, statistical distributions based on the biomarker parameters 120,and the cognitive phenomena data 124 can be generated. In some examples,the biomarker parameters 120 can be mapped to one or more probabilitydensity functions (PDFs), and tested for the ability of the biomarkerparameters 120 to predict the associated cognitive phenomena data 124.PDFs that can reliably predict the demonstrate the cognitive phenomenadata 124 based on the biomarker parameters 120 can be used to as thestatistical cognitive model 122. In even further examples, thestatistical cognitive model 122 can use multiple PDFs that can becombined via weighting or voting to predict a cognitive phenomenon,e.g., multiple biomarker parameters 120 can be used as input to predicta cognitive phenomenon. Moreover, a joint modeling framework can beprovided by combining the biomarker parameters 120 derived based on themultiple atlases to generate the statistical cognitive model 122.Additionally, the method 220 can be repeated (e.g., iterated) asadditional data becomes readily available. Accordingly, over time thepower of the statistical cognitive model 122 can be substantiallyimproved.

The statistical cognitive model 122 can be used to predict a cognitivephenomenon based on the biomarker parameters 120 of associated with thesubject 24. For example, an instance of the biomarker parameters 120according to a single test can be inputted to the statistical cognitivemodel 122. The statistical cognitive model 122 can output a probabilityaccording to the PDF associated with the statistical cognitive model122. The probability can provide a quantitative assessment of theinstance of the biomarker parameters 120. For example, an fMRI image canbe analyzed statistically by the statistical cognitive model 122 for aparticular cognitive disorder. Accordingly, a diagnosis can be providedalong with an estimate of the severity of the diagnosis.

In some examples, the statistical cognitive model 122 can quantify thedifference between normal and abnormal networks in the brain of thesubject 24. For example, the biomarker parameters 120 associated withthe subject 24 can be analyzed statistically by the statisticalcognitive model 122. The statistical cognitive model 122 can outputstatistics corresponding to the difference between the biomarkerparameters 120 of the subject 24 and the population used to generate thestatistical cognitive model 122. Accordingly, the gap between normalnetworks of the subject 24 and abnormal networks of the population canbe statistically quantified based on the biomarker parameters 120.Likewise, the gap between abnormal networks of the subject 24 and normalnetworks of the population can be statistically quantified based on thebiomarker parameters 120. Accordingly, the biomarker parameters 120 canbe used classify and quantify the result of network perturbationsassociated with cognitive phenomena such as, but not limited to, headtrauma (e.g., concussions), Post-Traumatic Stress Disorder (PTSD),schizophrenia, and Alzheimer's.

Referring to FIG. 1, the quantitative gap can be reported to a clinicianto assist with rapid clinical diagnosis of the degree of head trauma orother cognitive phenomena. For example, the subject 24 can be suspectedof being suffering from a cognitive deficit or cognitive disorder.Neural activity data 21 can be collected from the subject 24. The neuralactivity data 21 can be communicated over the network 12 to the neuralarchitecture system 100. The neural modeling engine 110 can beprogrammed to reduce the neural activity data 21 into the neural model112 and biomarker parameters 120 useful for diagnosing the cognitivedeficit or the cognitive disorder. As described herein, the biomarkerparameters 120 can then be evaluated with the statistical cognitivemodel 122 to classify the cognitive deficit or cognitive disorder andquantify the severity of the cognitive deficit or cognitive disorder.The neural model 112, biomarker parameters 120, the classification andquantification, or combinations thereof can be communicated to theclient device 108 associated with the clinician. Alternatively, oradditionally, the client device 108 can be configured to execute thestatistical cognitive model 122. Thus, the client device 108 can beprovided with the biomarker parameters 120 and classify the cognitivedeficit or cognitive disorder, and quantify the severity of thecognitive deficit or cognitive disorder.

Accordingly, the clinician can initiate a treatment plan(minimal-to-aggressive) based in part on severity of cognitive deficitor cognitive disorder. The use of the biomarker parameters 120 can beused to provide insight through the use of a large statisticalpopulation that is not available according to existing imagingmodalities (e.g. MRI). For example, while the neural activity data 21can be examined manually to reveal primarily anatomical changes, suchanatomical changes may not be instructive in identifying many cognitivephenomena to the clinician. There can be substantial cognitivedisruption in the absence of anatomical disruption (e.g. concussion).Moreover, the efficacy of the treatment plan can be monitored byperiodically collecting new biomarker parameters 120 for evaluation withthe statistical cognitive model 122. The classification andquantification of the cognitive deficit or cognitive disorder can trackincremental changes that cannot be done practically via direct review ofanatomical changes alone.

FIG. 8 depicts an example of a method 300 for analyzing a connectivitymatrix for generating a neural model (e.g., the neural model 112, asillustrated in FIG. 1). The connectivity matrix can be analyzed with aneural model engine (e.g., the neural modeling engine 110, asillustrated in FIG. 1). The method can begin at 305 wherein neuralactivity data 21 can be transformed to [0, 1] based on topologicalstatistics of a network associated with a brain of the subject 24. Priorto transformation of the neural activity data 21, the neutral activity21 can be parameterized, as described herein. At, 310, parameters thatmaximize Equation 5 can be determined. For example, θ and β parameterscan be evaluated such that Equation 5 is maximized (e.g., throughalternating maximization of θ and β parameters, as described herein).The θ parameters are parameter estimates corresponding to endogenousnetwork statistics.

The β parameters can be updated by applying Beta regression to estimateeffects of exogenous covariates. Thus, parameters of exogenouscovariates can be estimated. The θ parameters can be updated bysimulating networks based on current estimates (e.g., via MCMC) for aplurality of N iterations, wherein N is an integer greater than one.During each iteration, noise can be added to create proposed networks,which can be evaluated relative to prior networks (or current networks).Such an evaluation can include calculating statistics of both networksand accepting or rejecting a given proposed network. The simulatednetworks can correspond to the denominator in Equation 2. The θparameters can be identified that maximize a likelihood in Equation 5.At 315, a determination can be made if the θ parameters have changed bya given selected value. If the θ parameters have changed by the givenselected value, the method can proceed to step 305. However, if the θparameters have not changed by the given selected value, the method canproceed to 320. At 320, the neural model can be determined as converged,and results can be outputted. Accordingly, the neural model can begenerated.

EXAMPLE

Referring to FIGS. 1 and 3, an example of the neural modeling engine 110according to the neuro-GERGM was evaluated to confirm a model fit. Theexample included using neural activity data 21 collected from restingstate activity of a default mode network (DMN), and evaluated restingstate functional connectivity (rsFC). Neural activity data 21 wascollected from 250 subjects 24 that each completed a six-minute restingstate scan in which their only task was to stay awake. Since excessivemotion can introduce artificial correlations across regions inconnectivity analyses, a strict movement cut-off was employed, whichexcluded any participant who moved greater than 2 millimeters (mm) whileundergoing resting state. Further, some subjects 24 had their brainoutside of the full field of view, and as such were excluded. Together,the study yielded neural activity data 21 for 154 total analyzablesubjects 24.

Functional parcellation of the DMN was used to construct theconnectivity matrix ρ. Since the parcellation contained only structureson the midline and left hemisphere, corresponding regions on the righthemisphere were used. Additionally, since there are asymmetries in theDMN across hemispheres, we adjusted the hotspots in the right hemispherebased on coordinates from NeuroSynth. As a result, a 20-node atlas wasused.

The atlas was applied to the time-series fMRI data of each subject 24,such that each voxel within a region was averaged together, yielding 20time-series' for each subject 24. The connection weights between regionsof the connectivity matrix ρ was calculated with the Pearson correlationcoefficient between pairs of regions, partialling out activity relatedto cerebrospinal fluid, white matter, and motion. This yielded a 20×20symmetric matrices for each participant corresponding to the connectionstrength between each of the 20 regions. Finally, each participants'20-node atlas was averaged to yield a single connectivity matrix ρ(e.g., a 20×20 matrix) that corresponded to the average connectivityacross all subjects 24 in order to approximate an average brain. It isnoted that such averaging was selected for model validation, and astrength of the examples described herein is the ability of recoveringparameters for each individual, allowing for comparison acrossindividuals (as well as within individuals across tasks).

In this example, 15 different models were fitted to the DMN. Allstatistics included an edges statistic (which corresponds to anintercept), and the remaining statistics corresponded to all 15 possiblecombinations of four statistics. Two of the statistics—two-stars andtransitive triads—modeled endogenous statistics of the network. Theother two statistics are exogenous statistics: the spatial distancebetween regions, and an indicator specifying whether or not two regionsare in the same hemisphere of the brain (Hemisphere node match). T(·|β)was set as a Beta distribution, and the partial correlations weremodeled as a regression on the exogenous predictors using Betaregression. Table 1, which is provided below, describes the predictorsused in each of the models. To avoid model degeneracy, a geometric downweighting strategy was applied to the structural predictors. Thestrategy resulted in geometric down weights of 0.8 for two-stars and 0.4for transitive triads.

TABLE 1 Description of network statistics and model specifications.Structural Exogenous Predictor Description Model Model Edges Edgedensity of the

network Two - Stars Density of two-stars.

Measure of popularity of nodes Transitive Triads Density of triangles.

Measure of clustering Spatial distance Euclidean distance

between regions Hemisphere node Indicator specifying if two

match regions are in the same hemisphere

The neuro-GERGM output both a parameter estimate and a standard errorfor each statistic specified. The coefficient outputs of the statisticscan be found in Table 2. Across all statistics, a significantly negativetwo-stars coefficient, and a significantly positive transitive triadscoefficient was determined. Further, the statistics were relativelyunchanged with the addition of the exogenous covariates. Together, thissuggests that, at least for the present atlas, the DMN displays moretriadic closure (e.g., clustering) than would be expected by chance, butless preferential attachment (e.g., the use of hubs) than would beexpected by chance.

TABLE 2 Network output coefficients, numbers in parentheses correspondto standard errors. Transitive Edges Two-Stars Triads HemisphereDistance Endogenous .24(.02) −.44(.11) .22(.05) Exogenous .86(.10)−.45(.11) .21(.05) .06(.024) .08(.02)

To assess whether the structural features were recovered, t-tests wereconducted to determine whether the distribution mean was significantlydifferent from the observed value. In this example, higher p-values aredesired, as they suggest that the two means are not statisticallydifferent from one another (e.g., better fit). The endogenous statisticsproduced simulated networks that were statistically indistinguishablefor the two-stars statistic (p=0.99) and the network density (p=0.40)statistic, but not ttriads (p=0.02). The exogenous statistics, on theother hand, produced simulated networks that were statisticallyindistinguishable for all three statistics (two-stars: p=0.66; ttriads:p=0.47; density: p=0.30).

In another set of experiments, we have modeled additional subnetworkswithin the brain and found largely consistent findings in regards to theorganizational factors. In this investigation, we used a functionalatlas that provides subnetwork groupings for each of 264 brain regions.Of these networks, we modeled 9 different subnetworks, including aDorsal Attention Network, a Frontoparietal Control Network, a VentralAttention Network, a Visual Network, an Auditory Network, a RewardNetwork, a Subcortical Network, a Salience Network. The size of thesubnetworks range in size from 9 nodes to 31 nodes. Across all of thesesubnetworks, we find either a significantly negative two-stars statistic(or a directional but non-significant effect), and a significantlypositive triads statistic (or a directional but non-significant effect).

In another demonstration, we have recently applied the descriptivestatistics approach to investigating schizophrenia. Specifically, wetook a dataset with 145 individuals (both schizophrenics and healthycontrols) who underwent 6 minutes of resting state fMRI. For eachparticipant, we constructed a weighted correlation matrix, andcalculated the following weighted descriptive statistics: clustering,eigenvector centrality, betweenness centrality, closeness centrality,and degree centrality. We input these metrics into a machine-learningalgorithm, and developed a classifier that could predict with 75-85%cross-validation accuracy whether an individual was healthy orschizophrenic.

It should now be understood that the examples provided herein canconduct statistical inference and hypothesis testing on neurologicalnetwork data without departing from the basic structure of the data. Forinstance, the neuro-GERGM can be used to evaluate neural activity data(e.g., fMRI images, DTI images, or both) to generate parsimonious andwell-fitting models of neurological networks. Additionally, thedimensionality of the intractable problem of comparing neural activitydata across multiple subjects can be reduced from a large dimensionproblem (e.g., about 100K dimension) to a small dimension problem (e.g.,between about 3-8 dimensions). For example, the use of the biomarkerparameters for cross and intra network comparisons can substantiallyreduce the memory and processing requirements for modeling cognitivephenomena.

Additionally, the neuro-GERGM can quantify the intrinsic architecture ofneural networks in a manner that does not require the reduction orcensoring of data. For example, the neuro-GERGM can probe the functionalorganization of the brain to produce well-fitting models that capturethe structural properties of neurological networks, and predict featuresof an out of sample network with high accuracy.

It is noted that the terms “substantially” and “about” can be utilizedherein to represent the inherent degree of uncertainty that may beattributed to any quantitative comparison, value, measurement, or otherrepresentation. These terms are also utilized herein to represent thedegree by which a quantitative representation may vary from a statedreference without resulting in a change in the basic function of thesubject matter at issue.

What have been described above are examples. It is, of course, notpossible to describe every conceivable combination of components ormethods, but one of ordinary skill in the art will recognize that manyfurther combinations and permutations are possible. Accordingly, thedisclosure is intended to embrace all such alterations, modifications,and variations that fall within the scope of this application, includingthe appended claims. Additionally, where the disclosure or claims recite“a,” “an,” “a first,” or “another” element, or the equivalent thereof,it should be interpreted to include one or more than one such element,neither requiring nor excluding two or more such elements. As usedherein, the term “includes” means includes but not limited to, and theterm “including” means including but not limited to. The term “based on”means based at least in part on.

What is claimed is:
 1. A computer-implemented method for modeling neuralarchitecture comprising: identifying, by one or more processors, regionsof interest of a brain of a subject based on image data characterizingthe brain of the subject; mapping, by the one or more processors, theregions of interest to a connectivity matrix, wherein the connectivitymatrix is a weighted and directed or undirected network; applying, bythe one or more processors, a multivariate transformation to theconnectivity matrix to transform the connectivity matrix into a partialcorrelation matrix, wherein the multivariate transformation maintains apositive definite constraint for the connectivity matrix; andtransforming, by the one or more processors, the partial correlationmatrix into a neural model indicative of the connectivity matrix.
 2. Thecomputer-implemented method of claim 1, wherein the transforming thepartial correlation matrix into the neural model indicative of theconnectivity matrix is based on statistics determined according to theconnectivity matrix; further comprising, transforming, by the one ormore processors, the statistics into biomarker parameters.
 3. Thecomputer-implemented method of claim 2, wherein the statistics compriseendogenous statistics.
 4. The computer-implemented method of claim 3,wherein the endogenous statistics comprise one of two-stars, transitivetriads, intercept term, and any of a class of statistics computed assums of sub-network products or transformations thereof.
 5. Thecomputer-implemented method of claim 3, wherein the statistics furthercomprise exogenous statistics.
 6. The computer-implemented method ofclaim 1, further comprising scaling, by the one or more processors, theconnectivity matrix prior to the multivariate transformation.
 7. Thecomputer-implemented method of claim 1, further comprising determiningand mapping, by the one or more processors, the regions of interestaccording to an activity of the subject.
 8. The computer-implementedmethod of claim 1, wherein the regions of interest are determined andmapped according to an atlas associated with the subject.
 9. Thecomputer-implemented method of claim 8, wherein the atlas delineates oneof a whole brain network, Default Mode Network, a Dorsal AttentionNetwork, a Frontoparietal Control Network, a Ventral Attention Network,a Visual Network, an Auditory Network, a Reward Network, a SubcorticalNetwork, a Salience Network, and any other network or subnetwork thatcaptures functional or structural connectivity, and a combinationthereof.
 10. The computer-implemented method of claim 1, furthercomprising: identifying, by the one or more processors, distinct regionsof interest in the image data; and mapping, by the one or moreprocessors, the distinct regions of interest of the image data to theconnectivity matrix, wherein the regions of interest and the distinctregions of interest provide a joint modeling framework.
 11. Thecomputer-implemented method of claim 1, wherein transforming, by the oneor more processors, the partial correlation matrix into the neural modelindicative of the connectivity matrix is according to a descriptiveneural modeling engine, wherein values of the descriptive statisticsprovide an indication of a neurological condition and/or a relatedseverity of the neurological condition when compared against a referencedistribution of the descriptive statistics, wherein descriptivestatistic corresponds to a biomarker for the neurological condition. 12.The computer-implemented method of claim 1, wherein transforming thepartial correlation matrix into the neural model indicative of theconnectivity matrix is according to a neurological generalizedexponential random graph model (neuro-GERGM) algorithm.
 13. Thecomputer-implemented method of claim 1, wherein the image data comprisesone of magnetic resonance imaging (MRI) data, functional magneticresonance imaging (fMRI) image data, diffusion weighted imaging (DWI)data, diffusion tensor imaging (DTI) data, electromyogram (EMG) data,electroencephalogram (EEG) data, positron emission tomography (PET)data, magnetoencephalography (MEG) data, Electrocorticography (ECoG)data, ultrasound imaging data, and a combination thereof.